Scientific Research is a Teamwork on behalf and benefit of the entire Mankind


CfA

Peer-reviewed Academic Publications










We abridged the videos above by original imagery, with kind permission endorsed by dr. Gareth V. Williams, astronomer at the Solar, Stellar and Planetary Sciences Division of the Harvard-Smithsonian Centre for Astrophysics, and Associate Director of the Minor Planet Center, at Cambridge, Massachusetts.   Recommended an Internet bandwidth wide enough to stream the video in its native 1420×730 pixels High Definition (best vision in full screen).   Definition high enough to sight the tens of thousands of minor bodies, Comets and Asteroids, orbiting the inner and middle areas of the Solar System, drawn as seen from the North Ecliptic Pole with the vernal equinox off to the right.    At right side the enlarged detail of the inner Solar System.   The Sun represented as a yellow-coloured starlike symbol and blue-coloured the Earth.   A cloud of objects orbits the Solar System barycentre, between the orbits of Mars and Jupiter.  


       Further infos:           www.cfa.harvard.edu


Peer-reviewed Academic Publications





















asteroids-between-jupiter-2
We abridged the videos above by original imagery, with kind permission by dr. Gareth V. Williams, astronomer at the Solar, Stellar and Planetary Sciences Division of the Harvard-Smithsonian Centre for Astrophysics, and Associate Director of the Minor Planet Center, at Cambridge, Massachusetts.   Recommended an Internet bandwidth wide enough to stream the video in its native 1420×730 pixels High Definition (best vision in full screen).   Definition high enough to sight the tens of thousands of minor bodies, Comets and Asteroids, orbiting the inner and middle areas of the Solar System, drawn as seen from the North Ecliptic Pole with the vernal equinox off to the right.    At right side the enlarged detail of the inner Solar System.   The Sun represented as a yellow-coloured starlike symbol and blue-coloured the Earth.   A cloud of objects orbits the Solar System barycentre, between the orbits of Mars and Jupiter.  



Further infos:           www.cfa.harvard.edu



Listed below 4 peer-reviewed  academic publications, contributed by Roberto Alfano.  Juvenile experimental and computational studies of some celestial objects.  Three of them are integrated in the Minor Planet Circulars (MPC) since 1947 published by the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Massachusetts, USA.   

 The minor bodies of the Solar System orbiting between Mars and Jupiter in a 3D rendering by the Institut de Mécanique Céleste et de Calcul des éphémérides (IMCCE), Paris Observatory, France (  Berthier, et al./ Paris Observatory/2006) 















The Minor Planet Center,  operating under the auspices of the division F of the International Astronomical Union (IAU), is the global official scientific and professional body dealing with astrometric observations and orbits of comets and asteroids.   The International Astronomical Union is the largest body of professional astronomers in the world.   It is equivalent to other bodies like the IUPAC, International Union of Pure and Applied Chemistry and others existing for the Physical Sciences.   The Circulars contain astrometric observations, orbits and ephemerides of both asteroids and comets.   Also, new numberings and namings of minor planets, as well as numberings of periodic comets, are announced in the Circulars.

 The minor bodies of the Solar System orbiting between Mars and Jupiter in a 3D rendering by the Institut de Mécanique Céleste et de Calcul des éphémérides (IMCCE), Paris Observatory, France (  Berthier, et al./ Paris Observatory/2006) 








Listed below 4 peer-reviewed  academic publications, contributed by Roberto Alfano.  Juvenile experimental and computational studies of some celestial objects.  Three of them are integrated in the Minor Planet Circulars (MPC) since 1947 published by the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Massachusetts, USA.  The Minor Planet Center is the global official scientific and professional body dealing with astrometric observations and orbits of comets and asteroids.   It operates under the auspices of the division F of the International Astronomical Union (IAU),  The IAU is the largest body of professional astronomers in the world.   It is equivalent to other bodies like the IUPAC, International Union of Pure and Applied Chemistry and similar existing for the Physical Sciences.   The Circulars contain astrometric observations, orbits and ephemerides of both asteroids and comets.   Also, new numberings and namings of minor planets, as well as numberings of periodic comets, are announced in the Circulars.



Astrometric topocentric measurements of comet

Levy 1990K01 

published in:  

eds., Brian G. Marsden, Gareth V. Williams, Circulars, Harvard-Smithsonian Astrophysical Observatory, Minor Planet Center, at Cambridge, Mass., MPC 16929-17070, pages 16935-16936, October 4, 1990                                                 

141 pages, PDF, 340 KB          

Cross-references:    

  • DOI: 10.13140/2.1.4107.0889   
  • ISSN 0736-6884 
  • Dewey 528 C49m1a
  • OCLC 5622733         

Astronomical topocentric positions (α, δ, t) of the newly discovered comet Levy 1990 K01, obtained at the Genoa Astronomical Observatory.   Data reduction of the reference stars by mean of Runge-Kutta method.  Activities following the program outlined by Brian G. Marsden, Minor Planet Center, Harvard-Smithsonian Astrophysical Observatory, Cambridge, Mass.  Data integrated in the orbital parameters on pages 16935-16936.

Related subjects

Measurements of Physical Properties, Comets, Astrometry, Data Analysis, Error Analysis, Minor Planets, Stellar catalogues, Astrography, Harvard-Smithsonian Center for Astrophysics, Minor Planet Center, Comet Levy 1990 K01



Astrometric topocentric measurements of comet 

P/Brorsen-Metcalf 23P

published in:  

eds., Brian G. Marsden, Conrad M. BardwellCirculars, Harvard-Smithsonian Astrophysical Observatory, Minor Planet Center, at Cambridge, Mass., MPC 15121-15280, October 14, 1989   

159 pages, PDF, 340 KB   

Cross-references:  

  • DOI: 10.13140/2.1.5155.6645        
  • ISSN 0736-6884        
  • Dewey 528 C49m1a     
  • OCLC 5622733

Astronomical topocentric positions (α, δ, t) of the newly discovered comet P/Brorsen-Metcalf 23P, obtained at the Genoa Astronomical Observatory.    Data reduction of the reference stars by mean of Runge-Kutta method.  Activities following the program outlined by Brian G. Marsden, Minor Planet Center, Harvard-Smithsonian Astrophysical Observatory, Cambridge, Mass.  Data integrated in the orbital parameters on pages 15130-15133-15134.

Related subjects

Measurements of Physical Properties, Comets, Astrometry, Data Analysis, Error Analysis, Minor Planets, Stellar catalogues, Astrography, Harvard-Smithsonian Center for Astrophysics, Minor Planet Center, Comet P/Brorsen-Metcalf 23P



Astrometric topocentric  measurements of comet 

P/Austin 1989C1 

published in:  

eds., Brian G. Marsden, Conrad M. BardwellCirculars, Harvard-Smithsonian Astrophysical Observatory, Minor Planet Center, at Cambridge, Mass., MPC 16473-16636, July 8, 1990   

164 pages, PDF, 352 KB       

Cross-references:     

  • DOI: 10.13140/2.1.1485.6481   
  • ISSN 0736-6884            
  • Dewey 528 C49m1a
  • OCLC 5622733 

Astronomical topocentric positions (α, δ, t) of the newly discovered comet P/Austin 1989C1, obtained at the Genoa Astronomical Observatory.   Data reduction of the reference stars by mean of Runge-Kutta method.  Activities following the program outlined by Brian G. Marsden, Minor Planet Center, Harvard-Smithsonian Astrophysical Observatory, Cambridge, Mass.  Data integrated in the orbital parameters on pages 16473-16636.

Related subjects

Measurements of Physical Properties, Comets, Astrometry, Data Analysis, Error Analysis, Minor Planets, Stellar catalogues, Astrography, Harvard-Smithsonian Center for Astrophysics, Minor Planet Center, Comet P/Austin 1989C1




Photographic Astrometry Techniques 

(Astrometria Fotografica)

Genoa Astronomical Observatory, at Genoa, Italy, 1988        

132 pages, PDF, with 36 figures and 23 tables      

Cross-references and download:   

A comprehensive methodological approach to the theoretical and experimental deduction of equatorial astrometric topocentric positions of newly discovered comets and asteroids.  The research here presented shows the critical exam of the entire instrumental and data analysis methodology: it is not a mere digest by other publications.  Some of the subjects treated:







  • Differential corrections and solutions;
  • Data rejection;
  • Propagation of the error during calculations;
  • Software to automate the differential and matricial calculations;
  • The Runge-Kutta method of Numerical Analysis to minimise the residuals of the matricial computations; 
  • Format of the data for the Harvard-Smithsonian Centre for Astrophysics;
  • Artificial Intelligence-driven choice of the set of reference stars;
  • Guide Star Catalog (GSC), Smithsonian Astrophysical Observatory Catalog (SAO);
  • Stellar atlases;
  • Minor Planet Circulars
  • Positions, proper motions and radial velocities;
  • Astrographs for Astrometry;
  • Observations;
  • Plate measurements by Zeiss macro-micrometer;
  • Plate data reduction, 
  • Plate distortions, 
  • Time coordinate obtained by atomic clocks.

Brian Geoffrey Marsden Harvard-Smithsonian Center for Astrophyisics

A memory of the prof. Brian Geoffrey Marsden (1937-2010).  The video below, shot in 1989, epoch of our collaboration, when he was Director of the Harvard-Smithsonian Center for Astrophysics' Minor Planet Center

The Method of the Standard Coordinates has been used to provide astrometric topocentric positions for three newly discovered comets to the Minor Planet Center, whose residuals resulted < 1.2” (arcseconds).   The Author conducted the research when Director of the Genoa Astronomical Observatory using its optic equipments and facilities. Observatory for which obtained the code 974 by the Minor Planet Center of the International Astronomical Union, division F.    Cometary positions were then integrated by the Minor Planet Center (prof. Brian G. Marsden, visible in the video at right side, and astronomers Gareth V. Williams and Conrad M. Bardwell) into the respective sets of orbital parameters.   Then, published in the peer-reviewed Harvard-Smithsonian Center for Astrophysics’ Minor Planet Circulars (M.P.C.s) of the International Astronomical Union, since 1947 published under the ISSN 0736-6884.   

Peer-reviewed

The content of the book was peered-reviewed by the professional Astronomers: 

and awarded in the scientific contest held in Rozzano (Milan area, Italy) on May 1988.   

Related subjects

Measurements of Physical Properties, Optomechanics, Comets, Astrometry, Data Analysis, Error Analysis, Minor Planets,  Runge-Kutta method, Stellar catalogues, Astrography,  Standard coordinates, Proper Motions, Stellar Kinematics, Parallax, Radial Velocity, Harvard-Smithsonian Center for Astrophysics, Minor Planet Center 





The orbital elements of newly discovered comets and asteroids, collectively named Minor Bodies of the Solar System are always poorly defined, reason why they are prone to be lost.   Massive amounts of high quality positions, obtained by worldwide Astronomical Observatories equipped for astronomical positional research are then urgently needed by the scientists.  The International Astronomical Union’s  Division F, Minor Planet Center based at the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Massachusetts, USA.    This is the official body caring these aspects of Positional Astronomy on behalf of the global scientific research astronomical community (International Astronomical Union, IAU).  Also, some asteroid and comet discoveries of previous decades can be lost because not enough observational data had been obtained at the time to determine a reliable enough orbit.   Reliable enough to allow to know where to look for re-observation at future dates.  Sometimes, a newly discovered object turns out to be a rediscovery of a previously lost object, which can be determined by calculating its orbit backwards into the past and matching calculated positions with the previously recorded positions of the lost object.   In the case of comets this is especially complex because of nongravitational forces that can affect their orbits.  





Commercial rationale

The interest is not only scientific.  There are today several well funded initiatives devoted to the near future commercial exploitation of the impressive resources of minor bodies.  These are mainly mountain-like masses of high valued expensive metals like Platinum, Gold or Iridium.   The less lucky encounters being a mountain-like masses of nearly pure Iron or Manganese.  Planetary protection rationale

Nearly whoever remember the fearful astonishing footage arrived from Celiabinsk (Russia) last February 15, 2013.  An asteroid with extremely limited diameter, exploded around 20 km of altitude, leaving a destruction of windows which wounded ~1200 people.  The gravity of the effects left many in that military-industry oriented town imagine in the initial minutes it’d be an attack with a nuclear bomb detonated at high altitude, shortly before the followings at ground level.    An asteroid big enough to destroy a city strikes the Earth every 100 years.  The strategy of the entire Humanity for preventing disasters pass mainly thru the activity of Donald K. Yeomans, today the manager of NASA’s Near-Earth Object Program Office at the Jet Propulsion Laboratory in Pasadena, California, USA. 

The Near-Earth Objects Program Office at JPL, along with the Minor Planet Center of the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Mass., USA, helps coordinate the search for, and tracking of, asteroids and comets passing into Earth’s neighborhood, to identify possible hazards to Earth.      It says there is evidence of 26 explosions in the upper atmosphere between the year 2000 and 2013, ranging in energy from 1 to 600 kilotons.   Donald K. Yeomans visible in the video here on right side, is the same Astronomer who personally kindly provided to the Author of these notes, Roberto Alfano, academic texts published by NASA documenting the then most updated data reduction techniques available to the Astrometric community astronomers.   The video below, illustrates the frequency and power in kiloton of TNT of the recent encounters the Earth made with asteroids.









Laplace: 

how to deduce orbital elements from measurements







Astrometry is devoted to obtaine orbital parameters.  Three quantities, a, e, T plus three angles i, ω, Ω collectively define the position of the body and its orbit at any given time t.  They are deduced by the positions provided by the Astrometric Research.  In the special case of a parabolic orbit, different than the elliptic shown in the figure above, the semi major axis which is infinite is replaced by the perihelion distance q, which defines the size of the parabola.  In the figure, the object defined as “Celestial body” can be a planet, asteroid, comet or an artificial satellite
In this section, we’ll deepen the relation between Astrometric Research and Orbit Calculations enters in the conversion of the Astrometric celestial positions to an orbit, by mean of Laplace's method, by the name of the French mathematician and astronomer Pierre-Simon marquis de Laplace, who first derived it.   To define completely an orbit in the space, the quantities:



































          ( a,  e,  T,  i,  ω,  Ω ) 



 Three quantities, a, e, T plus three angles i, ω, Ω collectively define the position of the body and its orbit at any given time t.  They are deduced by the positions provided by the Astrometric Research.  In the special case of a parabolic orbit, different than the elliptic shown in the figure above, the semi major axis which is infinite is replaced by the perihelion distance q, which defines the size of the parabola.  In the figure, the object defined as “Celestial body” can be a planet, asteroid, comet or an artificial satellite





collectively termed orbital elements (see figure on right side) have to be known.  The first two parameters above specify the size and the shape of the orbit in its orbital plane, while the other three define the orientation of the orbit with respect to the ecliptic plane, say that one complanar to the orbits of the majority of othe planets.   Since, in general, six elements are required to specify completely the orbit in space, it follows that six independent quantities must be obtained by the observations.   A single observation gives only two quantities (α, δ), in terms of the angular coordinates named Right Ascension α and Declination δ of the body.   And that’s why total three different sets of observations are required to start to define its orbit.  The orbital elements are:

  a      length of the semi major axis, the orbital size; 
  e      eccentricity, the orbital shape;
   i      inclination.  Angle between the orbital plane and the plane of the ecliptic;
  Ω     longitude of the ascending node. Angle from the vernal equinox along the ecliptic plane to the point of intersection of orbital plane with the ecliptic plane, defining where the orbit pins;
  ω     argument of the perihelion.  Angle measured from the ascending node to the perihelion point;
  T      time.  Position of the body in its orbit at the time of perihelion passage, used as a reference time to define the position of the body at other times.
Along the following derivation, we'll assume the object subject to the only influence of the Sun’s gravitational field, thus excluding other relevant influences, e.g. by the most massive plantes.    Let, at any given time t, the heliocentric equatorial rectangular coordinates of the orbiting body with respect to the plane of the celestial equator, be denoted:

 for the comet   (x, y, z)
 for the Earth    (X, Y, Z)
All celestial positions, extremely close satellites for telecommunications, planets or extremely far galaxies, since centuries are represented by two coordinates.   Two angles named Right Ascension and Declination.  The figure maps in arcminutes these coordinates for the “Blue Straggler Stars” candidates imaged in the Sculptor and Fornax constellations.  The red concentric ellipses indicate tidal and core radii. The small circles (magenta on the web) indicate the area within the tidal radii of the globular clusters in Fornax (abridged by Mapelli, M. et al./2009)




  All celestial positions, extremely close satellites for telecommunications, planets or extremely far galaxies, since centuries are represented by two coordinates.   Two angles named Right Ascension and Declination.  The figure maps in arcminutes these coordinates for the “Blue Straggler Stars” candidates imaged in the Sculptor and Fornax constellations.  The red concentric ellipses indicate tidal and core radii. The small circles (magenta on the web) indicate the area within the tidal radii of the globular clusters in Fornax (abridged by Mapelli, M. et al./2009)







Let their heliocentric distances be r and R which are given by:









                              r2  =  x2 + y2 + z2

                              R2  =  X2 + Y2 + Z2



Then, with reference to the triangle below at right side, being:

very small the mass of the orbiting body;
M      mass of the Sun;
me      mass of the Earth;


the acceleration of the celestial body center of mass with respect to is reduced to:

                              d2x/dt2  =  - G M x / r3



and, the acceleration of the center of mass of the Earth, results:

                              d2X/dt2  =  - G X (M + me)/R3



At any instant of time the motion of an orbiting object, natural or artificial, depicts a different triangle.  Its sides and angles are the distances r, R, ρ and the angle θ.  Because of the fact there are > 2 bodies in the system,  the solution of this triangle cannot have the infinite precision of the purely geometric calculations. The future position of a celestial body, known on the base of a set of orbital elements, cannot be known with infinite precision.  As an example, imagine a system Sun-Earth-Asteroid, and let the Asteroid be orbiting the Sun with a period of revolution e.g. 1 year.  In general, after 1 year, the distances r, R, ρ and the angle θ shall be be observed different than what hinted by newtonian mechanics.  A tendence to lose celestial objects which can only be countered by a massive flow of new astrometric positions (α, δ) of high quality, refreshing the set of orbital parameters.  High quality meaning that the vectorial sum of the residuals along both axes (α, δ) has to be < 2” 
 At any instant of time the motion of an orbiting object, natural or artificial, depicts a different triangle.  Its sides and angles are the distances r, R, ρ and the angle θ.  Because of the fact there are > 2 bodies in the system,  the solution of this triangle cannot have the infinite precision of the purely geometric calculations. The future position of a celestial body, known on the base of a set of orbital elements, cannot be known with infinite precision.  As an example, imagine a system Sun-Earth-Asteroid, and let the Asteroid be orbiting the Sun with a period of revolution e.g. 1 year.  In general, after 1 year, the distances r, R, ρ and the angle θ shall be be observed different than what hinted by newtonian mechanics.  A tendence to lose celestial objects which can only be countered by a massive flow of new astrometric positions (α, δ) of high quality, refreshing the set of orbital parameters.  High quality meaning that the vectorial sum of the residuals along both axes (α, δ) has to be < 2” 



Let the celestial body's (artificial satellite, comet, asteroid, etc.) geocentric:

direction cosines of the be  l, m, n
distance be ρ   
then:

                               x  =  X  +   l ρ

                               y  =  Y  +  m ρ

                               z  =  Z  +  n ρ



















































































    O - C 

< 2 arcsecond











































The direction cosines l, m, n are given in terms of the astrometric topocentric position of the celestial body, Right Ascension α and Declination δ:

                               l    =  cos α  cos δ

                               m  =  cos δ  sin α 

                               n   =  sin δ



The last seven equations above are referred to the heliocentric equatorial rectangular coordinate X of the orbiting body (itself referred to the plane of the celestial equator) we have the second order differential equation:



d2ρ/dt2 l  +  2 dρ/dt  dl/dt  +  ρ d2l/dt2    =   - { GM (X + l ρ) / r3 }  +  G(M + me) X /R3 



and, following a similar argument, other two equations in Y and Z.   Being θ  the angle between R and ρ, for the sides of the triangle in the figure above on right side, it may be derived that:

                          r2  =  R2 +  ρ2 - 2 ρ R cos θ    



By the projection of R in the direction of ρ:

                                R cos θ   =   -(l X  +  m Y  + n Z)



Solving the equation immediately above and the 2nd order differential equation before with respect to r and ρ, we have finally the possibility to obtain the celestial body heliocentric coordinates (x, y, z).     Their velocity components ( d x/d t,  d y/d t,  d z/d t ) can be derived for x  and  dx/dt  by:

                                  x   =   X  +  l  ρ

                                d x/d t   =   d X/d t  +  ρ  d l/d t   +  l  d ρ/d t



and following similar equations also for y, dy/dt, z and dz/dt.  Knowing position and velocity of the comet, the orbital elements and hence the orbit can be determined from the method already discussed.  The actual computation involves a knowledge of the value l, m, n, X, Y, Z and their derivatives.  The values of X, Y, Z have ephemeridal origin.  From this data the values of the first derivative can be found out. The calculation of geocentric direction cosines, their first and second derivatives can be deduced from (at least) three observations of the celestial body.  These astrometric observation should ideally be equally spread along the time and however not excessively close each other.    Let the dates of observation be:                  

                                                  t1,  t2,  t3



The Right Ascension α and Declination δ for these three times are known. Here we will just show an approximate method of getting the first and second derivatives. In the actual practice, it is possible to use several other methods.   The average value of the first derivative of l for the time between t1 and t2 and between t3 and t2 is given by:

                              d l12/d t   =   ( l2  -  l1 ) / ( t2  -  t1 )

                              d l23/d t   =   ( l3  -  l2  ) / ( t3  -  t2 ) 



If the time intervals are similar, say if:

                                               t2  -  t1   ≈   t3  -  t2    



then the value of  d l/d t  at time t2 is approximately equal to:

                                    d l2/d t   =   ( d l12/d t  +  d l23/d t ) / 2 



and same way for the second derivatives:

                         d2 l2/d t2   =   ½ { d l23/d t  -  d l12/d t ) / ( t3  -  t1 ) }

 

Finally, they can be obtained similar relations for the first and second derivatives of m and n.    As known by analytic geometry, to define the equation of a conic section are necessary (at least) three of its points.   Transposing geometry to the dynamical celestial arena, the elements obtained from (at least) three sets of observations, define the initial orbit of the object.  To ameliorate the precision of the orbit integrated by the points (observations), more and more observations are necessary.  As a matter of fact, equations can be set up for the difference between the predicted (C, as Calculated) and the observed (O, as Observed) positions.   These can then be solved to get the corrections for the preliminary orbital elements, leading to a more accurate set of orbital parameters.  Such an orbit is called a definitive orbit.   The orbit calculated on the base of the six elements gives the position in the space of the body (asteroid, comet, satellite or planet).     The spatial position of the Earth, referred to Solar System's centre of mass, a point closely lying but not coincident with the Sun's centre of mass, does matter if we want to determine the body's celestial position.   The geometrical position of the Earth can be obtained from the ephemeris, tables used to find the position of the comet in the plane of the sky.    In the special case of comets, the orbit gets perturbed due to the other masses as comets enter the Solar System, and mainly by the most massive planets, Jupiter and Saturn.   When the comet is very distant by the Solar System's centre of mass, also the orbital perturbation caused by the other stars, has to be considered.

  The orbits of all of the celestial bodies, close artificial satellites or comets some light years afar, owe their existence to Astrometric Research



















The orbital elements of newly discovered comets and asteroids, collectively named Minor Bodies of the Solar System are always poorly defined, reason why they are prone to be lost.   Massive amounts of high quality positions, obtained by worldwide Astronomical Observatories equipped for astronomical positional research are then urgently needed by the scientists.   The International Astronomical Union’s  Division F, Minor Planet Center based at the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Massachusetts, USA.    This is the official body caring these aspects of Positional Astronomy on behalf of the global scientific research astronomical community (International Astronomical Union, IAU).   Also, some asteroid and comet discoveries of previous decades can be lost because not enough observational data had been obtained at the time to determine a reliable enough orbit.   Reliable enough to allow to know where to look for re-observation at future dates.  Sometimes, a newly discovered object turns out to be a rediscovery of a previously lost object, which can be determined by calculating its orbit backwards into the past and matching calculated positions with the previously recorded positions of the lost object.   In the case of comets this is especially complex because of nongravitational forces that can affect their orbits.  



Commercial rationale

The interest is not only scientific.  There are today several well funded initiatives devoted to the near future commercial exploitation of the impressive resources of minor bodies.  These are mainly mountain-like masses of high valued expensive metals like Platinum, Gold or Iridium.   The less lucky encounters being mountain-like masses of nearly pure Iron or Manganese.




Donald K. Yeomans                 Jet Propulsion Laboratory 

I am indebted with prof. Donald K. Yeomans, for the assistance he granted to the astrometric researches here shown, providing Jet Propulsion Laboratory/NASA's original texts, otherwise impossible to encounter.   Donald K. Yeomans is today the manager of NASA’s Near-Earth Object Program Office at the Jet Propulsion Laboratory in Pasadena, California, USA. The Near-Earth Objects Program Office at JPL, along with the Minor Planet Center of the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Mass., USA, helps coordinate the search for, and tracking of, asteroids and comets passing into Earth’s neighborhood, to identify possible hazards to Earth.

Planetary protection rationale

Nearly whoever remember the fearful astonishing footage arrived from Celiabinsk (Russia) last February 15, 2013.   An asteroid with extremely limited diameter, exploded around 20 km of altitude, leaving a destruction of windows which wounded ~1200 people.  The gravity of the effects left many in that military-industry oriented town imagine in the initial minutes it’d be an attack with a nuclear bomb detonated at high altitude, shortly before the followings at ground level.   An asteroid big enough to destroy a city strikes the Earth every 100 years.  The strategy of the entire Humanity for preventing disasters pass mainly thru the activity of Donald K. Yeomans, today the manager of NASA’s Near-Earth Object Program Office at the Jet Propulsion Laboratory at Pasadena, California, USA. 


Energy released after different Events






   Energy released after different Events.  A meteorite of mass 1 kilotons striking the Earth at escape velocity, releases an energy nearly equivalent to a 17 kilotons implosion-type nuclear device (  J.A. Wheeler/1962) 


The Near-Earth Objects Program Office at JPL, along with the Minor Planet Center of the Harvard-Smithsonian Center for Astrophysics, at Cambridge, Mass., USA, helps coordinate the search for, and tracking of, asteroids and comets passing into Earth’s neighborhood, to identify possible hazards to Earth.   There is evidence of 26 explosions in the upper atmosphere between the year 2000 and 2013, ranging in energy from 1 to 600 kilotons.    Donald K. Yeomans visible in the video at right side, is the same Astronomer who personally kindly provided to the Author of these notes, Roberto Alfano, academic texts published by NASA documenting astrometric methods and the data reduction techniques.     



Laplace: 


how to deduce orbital elements from measurements




In this section, we’ll deepen the relation between Astrometric Research and Orbit Calculations enters in the conversion of the Astrometric celestial positions to an orbit, by mean of Laplace's method, name after the great French mathematician and astronomer Pierre-Simon marquis de Laplace, who first derived it.   To define completely the orbit in the space of a body (artificial or natural), the quantities:


     ( a,  e,  T,  i,  ω,  Ω ) 





















 Three quantities, aeT plus three angles i, ω, Ω collectively define the position of the body and its orbit at any given time t.  They are deduced by the positions provided by the Astrometric Research.  In the special case of a parabolic orbit, different than the elliptic shown in the figure above, the semi major axis which is infinite is replaced by the perihelion distance q, which defines the size of the parabola.  In the figure, the object defined as “Celestial body” can be a planet, asteroid, comet or an artificial satellite


collectively termed orbital elements (see figure above) have to be known.  The first two parameters above specify the size and the shape of the orbit in its orbital plane, while the other three define the orientation of the orbit with respect to the ecliptic plane, say that one complanar to the orbits of the majority of othe planets.   Since, in general, six elements are required to specify completely the orbit in space, it follows that six independent quantities must be obtained by the observations.   A single observation gives only two quantities (α, δ), in terms of the angular coordinates named Right Ascension α and Declination δ of the body.   And that’s why total three different sets of observations are required to start to define its orbit.  The orbital elements are:

  1.   a      length of the semi major axis, the orbital size; 
  2.   e      eccentricity, the orbital shape;
  3.    i      inclination.  Angle between the orbital plane and the plane of the ecliptic;
  4.   Ω     longitude of the ascending node. Angle from the vernal equinox along the ecliptic plane to the point of intersection of orbital plane with the ecliptic plane, defining where the orbit pins;
  5.   ω     argument of the perihelion. Angle measured from the ascending node to the perihelion point;
  6.   T      time.  Position of the body in its orbit at the time of perihelion passage, used as a reference time to define the position of the body at other times.

Along the following derivation, we'll assume the object subject to the only influence of the Sun’s gravitational field, thus excluding other relevant influences, e.g. by the most massive plantes.   Let, at any given time t, the heliocentric equatorial rectangular coordinates of the orbiting body with respect to the plane of the celestial equator, be denoted:

  •  for the comet   (x, y, z)
  •  for the Earth    (X, Y, Z)




  All celestial positions, extremely close satellites for telecommunications, planets or extremely far galaxies, since centuries are represented by two coordinates.   Two angles named Right Ascension and Declination.  The figure maps in arcminutes these coordinates for the “Blue Straggler Stars” candidates imaged in the Sculptor and Fornax constellations.  The red concentric ellipses indicate tidal and core radii. The small circles (magenta on the web) indicate the area within the tidal radii of the globular clusters in Fornax (  abridged by Mapelli, M. et al./2009)



Let their heliocentric distances be r and R which are given by:





                              r2  =  x2 + y2 + z2

                              R2  =  X2 + Y2 + Z2


then, with reference to the triangle below at right side, being:

  • very small the mass of the orbiting body;
  • M      mass of the Sun;
  • me      mass of the Earth;


the acceleration of the celestial body center of mass with respect to is reduced to:

                              d2x/dt2  =  - G M x / r3


and, the acceleration of the center of mass of the Earth, results:

                              d2X/dt2  =  - G X (M + me)/R3


 At any instant of time the motion of an orbiting object, natural or artificial, depicts a different triangle.  Its sides and angles are the distances r, R, ρ and the angle θ.  Because of the fact there are > 2 bodies in the system,  the solution of this triangle cannot have the infinite precision of the purely geometric calculations. The future position of a celestial body, known on the base of a set of orbital elements, cannot be known with infinite precision.  As an example, imagine a system Sun-Earth-Asteroid, and let the Asteroid be orbiting the Sun with a period of revolution e.g. 1 year.  In general, after 1 year, the distances r, R, ρ and the angle θ shall be be observed different than what hinted by newtonian mechanics.  A tendence to lose celestial objects which can only be countered by a massive flow of new astrometric positions (α, δ) of high quality, refreshing the set of orbital parameters.  High quality meaning that the vectorial sum of the residuals along both axes (α, δ) has to be < 2” 


Let the celestial body's (artificial satellite, comet, asteroid, etc.) geocentric:

  • direction cosines of the be  l, m, n
  • distance be ρ   

then:                       x  =  X  +   l ρ

                               y  =  Y  +  m ρ

                               z  =  Z  +  n ρ










The direction cosines l, m, n are given in terms of the astrometric topocentric position of the celestial body, Right Ascension α and Declination δ:

                                l   =  cos α  cos δ

                               m  =  cos δ  sin α 

                               n   =  sin δ


The last seven equations above are referred to the heliocentric equatorial rectangular coordinate X of the orbiting body (itself referred to the plane of the celestial equator) we have the second order differential equation:

   d2ρ/dt2 l  +  2 dρ/dt  dl/dt  +  ρ d2l/dt2    =   - [GM (X + l ρ) / r3] +  G(M + me) X /R3 


and, following a similar argument, other two equations in Y and Z.   Being θ the angle between R and ρ, for the sides of the triangle in the figure above on right side, it may be derived that:

                   r2  =  R2 +  ρ2 - 2 ρ R cos θ    


By the projection of R in the direction of ρ:

                     R cos θ   =  -(l X  +  m Y + n Z)


  The comets, speeding ~80000 kilometers-per-hour (> 22 km/s), are the fastest known material objects, i.e., ~3 times faster than ICBMs in the space


















Solving the equation immediately above and the 2nd order differential equation before with respect to r and ρ, we have finally the possibility to obtain the celestial body heliocentric coordinates (x, y, z).     Their velocity components ( d x/d t,  d y/d t,  d z/d t ) can be derived for x and dx/dt  by:

                 x   =   X  +  l  ρ

                dx/d t   =   dX/dt  +  ρ dl/dt   +  l dρ/dt


and following similar equations also for y, dy/dt, z and dz/dt.  Knowing position and velocity of the comet, the orbital elements and hence the orbit can be determined from the method already discussed.   The actual computation involves a knowledge of the value l, m, n, X, Y, Z and their derivatives.   The values of X, Y, Z have ephemeridal origin.  From this data the values of the first derivative can be found out.   The calculation of geocentric direction cosines, their first and second derivatives can be deduced from (at least) three observations of the celestial body.  These astrometric observation should ideally be equally spread along the time and however not excessively close each other.    Let the dates of observation be:                  

                                                      t1,  t2,  t3


The Right Ascension α and Declination δ for these three times are known. Here we will just show an approximate method of getting the first and second derivatives. In the actual practice, it is possible to use several other methods.   The average value of the first derivative of l for the time between t1 and t2 and between t3 and t2 is given by:

                                     dl12/dt   =   ( l2  -  l1 / ( t2  -  t)

                                     dl23/dt   =   ( l3  -  l2  / ( t3  -  t2 ) 


If the time intervals are similar, say if:

                                               t2  -  t     t3  -  t2    


then the value of  d l/d t  at time t2 is approximately equal to:

                                    dl2/dt   =   ( dl12/dt  +  dl23/dt ) / 2 


and same way for the second derivatives:

                         d2l2/dt2   =   ½ [ dl23/dt  -  dl12/dt ) / ( t3  -  t) ]

Horse Head Nebula in Orion. Astrometric Photographic Techniques similar to those hinted before are not only used to discover and track artificial satellites military and civil, comets and asteroids.  As an example, also many of the over 300 planets of other stellar systems discovered during past thirty years had been discovered also thanks to similar methods and equations. As known by analytic geometry, to define the equation of a conic section are necessary (at least) three of its points.   Transposing geometry to the dynamical celestial arena, the elements obtained from (at least) three sets of observations, define the initial orbit of the object.   To ameliorate the precision of the orbit integrated by the points (observations), more and more observations are necessary.   As a matter of fact, equations can be set up for the difference between the predicted (C, as Calculated) and the observed (O, as Observed) positions.   These can then be solved to get the corrections for the preliminary orbital elements, leading to a more accurate set of orbital parameters.  Such an orbit is called a definitive orbit.   The orbit calculated on the base of the six elements gives the position in the space of the body (asteroid, comet, satellite or planet).     The spatial position of the Earth, referred to Solar System's centre of mass, a point closely lying but not coincident with the Sun's centre of mass, does matter if we want to determine the body's celestial position.   The geometrical position of the Earth can be obtained from the ephemeris, tables used to find the position of the comet in the plane of the sky.    In the special case of comets, the orbit gets perturbed due to the other masses as comets enter the Solar System, and mainly by the most massive planets, Jupiter and Saturn.   When the comet is very distant by the Solar System's centre of mass, also the orbital perturbation caused by the other stars, has to be considered.

Astrometric Photographic Techniques, similar to those hinted before, are not only used to discover and track artificial satellites military and civil, comets and asteroids.  As an example, also many of the over 300 planets of other stellar systems discovered during past thirty years had been discovered applying associated methods







   O - C 

< 2 arcsecond





Finally, they can be obtained similar relations for the first and second derivatives of m and n.    As known by analytic geometry, to define the equation of a conic section are necessary (at least) three of its points.   Transposing geometry to the dynamical celestial arena, the elements obtained from (at least) three sets of observations, define the initial orbit of the object.   To ameliorate the precision of the orbit integrated by the points (observations), more and more observations are necessary.   As a matter of fact, equations can be set up for the difference between the predicted (C, as Calculated) and the observed (O, as Observed) positions.   These can then be solved to get the corrections for the preliminary orbital elements, leading to a more accurate set of orbital parameters.  Such an orbit is called a definitive orbit.   The orbit calculated on the base of the six elements gives the position in the space of the body (asteroid, comet, satellite or planet).     The spatial position of the Earth, referred to Solar System's centre of mass, a point closely lying but not coincident with the Sun's centre of mass, does matter if we want to determine the body's celestial position.   The geometrical position of the Earth can be obtained from the ephemeris, tables used to find the position of the comet in the plane of the sky.    In the special case of comets, the orbit gets perturbed due to the other masses as comets enter the Solar System, and mainly by the most massive planets, Jupiter and Saturn.   When the comet is very distant by the Solar System's centre of mass, also the orbital perturbation caused by the other stars, has to be considered.    






                                                                                                                                                                                                                                                                                                                                                                                                                                                         
Webutation
                                                                                                                       © 2013-2015 Graphene.  All rights reserved                                                         DMCA.com Protection Status                    

                                     
                                              
TRUSTe Privacy Policy Privacy Policy
Site protected by 6Scan